Q. 16 4.4( 7 Votes )

Let f be a function defined on [a, b] such that f ′(x) > 0, for all x (a, b). Then prove that f is an increasing function on (a, b).

Answer :

Since, f’(x) > 0 on (a,b)

Then, f is a differentiating function (a,b)

Also, every differentiable function is continuous,

Therefore, f is continuous on [a,b]

Let x1, x2 ϵ (a,b) and x2 > x1 then by LMV theorem, there exists c ϵ (a,b) s.t.

f’(c) =

Therefore, f is an increasing function.

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